# Is it possible to eliminate $g(x)$ in the formula $\dfrac{\int g(x)a(x)}{\int g(x)b(x)}$

I am hoping someone can show me how to do the following:

I have two integrals containing a function $g(x)$ that I would like
to eliminate.

The formula looks like this.

$\frac{\int g(x)a(x)}{\int g(x)b(x)}$

Is there any way to eliminate the $g(x)$ terms?

Thanks
Paul

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Not in general. – Raskolnikov Jul 13 '13 at 6:39
Probably more like hardly ever. – André Nicolas Jul 13 '13 at 6:46
People denote integrals in many different ways but the one used in this question is not one of them (you might want to add $dx$ at some well chosen places). – Did Jul 13 '13 at 8:19

Here's a simple counter example: \begin{align} g(x) &= x^k\\ a(x) &= x^m\\ b(x) &= x^n \end{align} Now $$\frac{\int g(x)a(x)\,\mathrm dx}{\int g(x)b(x)\,\mathrm dx} = \frac{(k+m+1)^{-1}x^{k+m+1}+C_1}{(k+n+1)^{-1}x^{k+n+1}+C_2}$$ but $$\frac{\int a(x)\,\mathrm dx}{\int b(x)\,\mathrm dx} = \frac{(m+1)^{-1}x^{m+1}+C_1}{(n+1)^{-1}x^{n+1}+C_2}$$ As you can see, not only does the presence of $g(x)$ make a difference, but the result also depends on the specifics of $g(x)$, like the value of $k$.
Thanks for the response. I forgot to mention that I can change the value for $a(x)$ and $b(x)$, so long as they do not contain $g(x)$ terms if that makes sense? Sorry for not mentioning that! – Paul Jul 15 '13 at 6:59